I am planning an approx. 10'x12' shed on level ground in south Texas (no freeze limits). I would prefer to use 2x4 joists for the floor framing, and 3/4" plywood for the flooring. How could I space supporting blocks under a 2x4 frame to ensure that it is stable and strong enough for everyday use? I.E. If I drop the spacing between blocks to 3' or even 2', would the 2x4 frame be strong enough to support general shed use (push lawnmowers, garden tools, bikes, etc)??

We do not recommend building an entire support frame out of 2x4's. They just do not offer the support you get when using 2x6 lumber. For more on this subject take a look at our support section, you may also find this link helpful www.deckplans.com/support/faq/buildingmaterials.html#support

Why do you want to use 2x4 for the floor joists? Even if you did add more blocks it will cost a LOT more for the extra blocks over the cost of 2x6.