I've seen another question (What's the most cost effective and efficient way to buy bricks?) that asks where the best place to buy specific bricks in small quantities are. That is helpful when working on LEGO projects that require precise pieces, but I am planning on using LEGO for non-artistic projects, such as building cases for Arduino that I have. As such, I am not really bothered by specific color. What is the most cost effective way to get common bricks in large bulk quantities such as the 4x1 or 2x4 bricks in any color?
The best and most cost effective way to get large quantities of brick, especially common ones, is via the LUGBulk program. You would need to be member of a LUG (LEGO User Group) that participates, and LUGBulk orders are due at specific times of the year. Search for a LUG near you and check.
Alternatively, you can go to your closest LEGO Store and see which common bricks they have in their Pick-a-Brick wall. If one of them is a piece you need, check with a store employee if they are willing to sell you a whole box. This used to cost $70 per box (independent of the type of brick inside) but I believe it might now be more ($100 IIRC).