Biblical references...
Leviticus 6:14-18... “And this is the law of the grain offering.  The sons of Aaron shall offer it before the Lord in front of the altar.  And one shall take from it a handful of the fine flour of the grain offering and its oil and all the frankincense that is on the grain offering and burn this as its memorial portion on the altar, a pleasing aroma to the Lord.  And the rest of it Aaron and his sons shall eat.  It shall be eaten unleavened in a holy place.  In the court of the tent of meeting they shall eat it. 
Deuteronomy 8:10... And you shall eat and be full, and you shall bless the Lord your God for the good land he has given you.
1st Samuel 16:7... But the Lord said to Samuel, “Do not look on his appearance or on the height of his stature, because I have rejected him. For the Lord sees not as man sees: man looks on the outward appearance, but the Lord looks on the heart.”
Psalms 100... (A Psalm for giving thanks).  Make a joyful noise to the Lord, all the earth!  Serve the Lord with gladness! Come into his presence with singing!  Know that the Lord, he is God!  It is he who made us, and we are his; we are his people, and the sheep of his pasture.  Enter his gates with thanksgiving, and his courts with praise.  Give thanks to him; bless his name!  For the Lord is good; his steadfast love endures forever, and his faithfulness to all generations.
Matthew 14:19... Then He ordered the crowds to sit down on the grass, and taking the five loaves and the two fish, He looked up to heaven and said a blessing.  Then he broke the loaves and gave them to the disciples, and the disciples gave them to the crowds. 
Matthew 15:36... He took the seven loaves and the fish, and having given thanks He broke them and gave them to the disciples, and the disciples gave them to the crowds.
Matthew 26:26... Now as they were eating, Jesus took bread, and after blessing it broke it and gave it to the disciples, and said, “Take, eat; this is my body.”
Mark 8:6... And He directed the crowd to sit down on the ground.  And He took the seven loaves, and having given thanks, He broke them and gave them to his disciples to set before the people; and they set them before the crowd.
Mark 8:7... And they had a few small fish.  And having blessed them, He said that these also should be set before them.
Mark 14:22... And as they were eating, He took bread, and after blessing it broke it and gave it to them, and said, “Take; this is my body.” 
Mark 14:23... And He took a cup, and when He had given thanks He gave it to them, and they all drank of it. 
Luke 18:10-14... “Two men went up into the temple to pray, one a Pharisee and the other a tax collector.  The Pharisee, standing by himself, prayed thus: ‘God, I thank you that I am not like other men, extortioners, unjust, adulterers, or even like this tax collector.  I fast twice a week; I give tithes of all that I get.’  But the tax collector, standing far off, would not even lift up his eyes to heaven, but beat his breast, saying, ‘God, be merciful to me, a sinner!’  I tell you, this man went down to his house justified, rather than the other.  For everyone who exalts himself will be humbled, but the one who humbles himself will be exalted.”
Luke 22:19... And He took bread, and when He had given thanks, He broke it and gave it to them, saying, “This is my body, which is given for you. Do this in remembrance of Me.”
Luke 24:30... When he was at table with them, he took the bread and blessed and broke it and gave it to them.
John 6:11... Jesus then took the loaves, and when He had given thanks, He distributed them to those who were seated.  So also the fish, as much as they wanted.
John 17:1... When Jesus had spoken these words, he lifted up his eyes to heaven, and said...
Acts 2:42... And they devoted themselves to the apostles’ teaching and the fellowship, to the breaking of bread and the prayers. 
Acts 27:35... And when he had said these things, he took bread, and giving thanks to God in the presence of all he broke it and began to eat. 
Romans 14:6... The one who observes the day, observes it in honor of the Lord.  The one who eats, eats in honor of the Lord, since he gives thanks to God, while the one who abstains, abstains in honor of the Lord and gives thanks to God. 
1st Corinthians 11 :23-24... For I received from the Lord what I also delivered to you, that the Lord Jesus on the night when he was betrayed took bread, and when He had given thanks, He broke it, and said, “This is my body, which is for you.  Do this in remembrance of me.”
Ephesians 5:20... giving thanks always and for everything to God the Father in the name of our Lord Jesus Christ.
1st Thessalonians 5:16-18... Rejoice always, pray without ceasing, give thanks in all circumstances; for this is the will of God in Christ Jesus for you. 
1st Timothy 4:4-5... For everything created by God is good, and nothing is to be rejected if it is received with thanksgiving, for it is made holy by the word of God and prayer.
James 4:17... So whoever knows the right thing to do and fails to do it, for him it is sin.

Discussion Thoughts...
Is it a sin to not pray before we eat?  The bible does not discuss the subject... nowhere in the bible is there any specific text that says we must pray before eating... neither is there any text that says "not praying before eating" is a sin!  However, in James 4:17 it says if we know that praying in the right thing to do and we do not pray it is a sin to not pray!  And in 1st Thessalonians 5:16-18 it says "Pray without ceasing, give thanks in all circumstances"... certainly giving thanks for the food we eat is a very logical and biblical thing to do!!!  there are numerous instances in the bible (see the list above all the way back to Leviticus) where people gave thanks for the food they ate. The first instance of Jesus blessing the food was in Matthew 14:19 where Jesus feeds the 5000... He looked up to heaven and said a blessing!  Again in Matthew 15:36 where he fed the 4000 He gave thanks before eating!  In Matthew 26:26 during the last supper, it is a bit different... as they were eating, Jesus took bread, and after blessing it... (in this instance he blessed the food as they were eating).  Mark tells about the last supper a bit differently... Jesus blessed the bread and separately blessed the cup!  In Acts 27:35 when Paul was shipwrecked he took some bread and blessed it before they ate.

Statistics...  Saying grace is a mostly a US tradition that is not predominate in other countries, not even in the countries that are traditionally considered mostly Catholic.  A poll by The Washington Post and the Kaiser Family Foundation finds that saying grace by American adults, 48% say they give blessings to God or say grace before meals at least a few times each week.  Slim 51% majorities say grace in both rural and urban America; in the suburbs, 45% say grace regularly.  62% of Republicans say grace at least a few times a week, compared with 43% of Democrats and 41% of independents.  60% Protestants say grace a few times a week or more, as do 52% of Catholics.  But the practice is more prevalent among black Protestants (80%) and white evangelical Protestants (74%) than among white mainline or nonevangelical Protestants, 31% of whom report saying grace frequently before meals.  Overall, 80% blacks, 60% Hispanics and 40% whites say grace at least a few times each week.

Different religions have different beliefs:
Protestantism... A typical Protestant prayer of grace written by John Wesley is "Be present at our table, Lord, Be here and everywhere adored.  Thy people bless, and grant that we may feast in paradise with thee.  Amen."
Catholics... A typical prayer before meals offered by Catholics is "Bless us, O Lord, and these, Thy gifts, which we are about to receive from Thy bounty.  Through Christ, our Lord. Amen."
Jewish... According to Jewish tradition they pray after they eat... this is based on Deuteronomy 8:10 listed above.
Islam... Before any meal, Muslims recognize that all of their blessings come from Allah.  Around the world, Muslims say a personal prayer (Du’a) both before and after their meals.
The Church of Latter-day Saints... In the LDS Family Guidebook Manual states: “Parents should see that family members learn to thank God for their food and ask Him to bless it before they eat.  Each person, including young children, should take a turn offering the blessing.  Offering a prayer for a blessing on the food helps parents and children learn to be thankful to our Heavenly Father.”

My Beliefs... This was another fascinating study... after reading and studying the various scriptures discussed above it is difficult to arrive at definitive conclusions!!!  Different scriptures point out that prayers were made before, during and-or after eating!  The bible is moot on when one should pray, but if we follow the example of Jesus, He prayed before (in one case during) eating... good examples to follow!!!  And, as it says in 1st Thessalonians 5:16-18 "pray without ceasing, give thanks in all circumstances"... certainly giving thanks for the food we eat is a "logical" circumstance!  James 4:17 is not specifically about prayers about eating, but is an all encompassing scripture about not praying is a sin if we know we should pray!  The bible is also moot on how to pray and how long to pray... there is no secret formula for prayer... you can bow, kneel, stand, or walk... God will hear you however you pray... in John 17:1 Jesus looked toward heaven when he prayed... you can pray silently or out loud...praying silently sometimes allows more focus on God... some people say is is easier to hear God if they are not doing all the talking!  Personal note... I pray a silent prayer before every meal... if I forget... I pray a silent prayer during eating or after!

Other studies... see a related study on prayer (scheduled for June release)... there are many other scriptures on prayer... this study dealt only with prayer at meals.