Woodpecker - 10 
Smash 5000 wooden blocks
This will most likely be the first block related achievement you earn as wood is probably the most common material in the game and can be broken by most birds. A good level to boost this on is Level 1-19. Just fire the fire bomb bird to destroy the most wood blocks as possible. Detonate it when you think your are in the best position. Pause and restart the level after the dust settles on your first shot. You can earn around 10 blocks per attempt of the level, netting around 100 per minute. It would take about 45 minutes to grind all 5000 wooden blocks.
Stone Cutter - 10
Smash 5000 stone blocks
Stone blocks are the grey ones. The best level to grind these on is Level 1-14. Use the three bomb birds to explode near the center of the three stone block clusters. You can get 15-25 blocks destroyed every attempt of the level, netting around 100 per minute. It would take about 45 minutes to grind all 5000 stone blocks.
Icepicker - 10
Smash 5000 iced blocks
Iced blocks are those that look clear and glass like. The best level to grind these on is Level 1-8. You get 4 birds that are made for destroying ice and nearly all ice blocks to break. Make sure you tap to split the bird into three before it makes contact to increase your effectiveness. Pause and restart the level to repeat when there is not many iced blocks left. You can get 15-25 blocks destroyed every attempt of the level, netting around 100 per minute. It would take about 45 minutes to grind all 5000 iced blocks. Level 1-11 may be better, but you get less splitting birds to work with.
Block Smasher - 10
Smash 50 000 blocks
See "Smash Maniac."
Smash Maniac - 10
Smash 500 000 blocks
These achievements are related to destroying any of the blocks in the game. Anything that you can destroy counts as a block. After finishing all other achievements, it will take you another 50 hours or more to smash a total of 500,000 blocks. If you are going to attempt this achievement, there are a few levels that work well.
Level 5-19: Use the three birds you are given to destroy the three large piles of miscellaneous blocks. I found hitting the bottom pile first, then the top, followed by a shot aimed towards the far right of the screen to clean up the mess was best. You can get around 40-60 blocks per attempt of the level, netting around 150 per minute. At this pace, it will take you around 55 hours to grind all 500,000 blocks.
Level 4-13: This level can also net you up to 80 blocks or so per attempt of the level. However, it is highly variable and a little trickier to make the shots necessary. The most important part is to hit the trees. They can be completely destroyed with one hit and contain around 15 blocks each. This outcome of this level is highly variable and you should probably just stick with the Level 5-19.
Level 1-16: Launch the two explosive birds at the blocks, one around the bottom and one around the top. You can get around 40-50 blocks per attempt of the level, earning around 150 blocks per minute. Thanks to PlanetGalida for suggesting this alternative level.
GUIDE UPDATES:
11/29/12 - Guide created and complete.
12/5/12 - Added another potential level to grind blocks for "Smash Maniac" based on a tip from PlanetGalida.
3/14/13 - Added an easier method to earning the "Big Spook" achievement. Thanks to zszaiss for the method.
3/18/13 - Fixed a few minor typos.