How does filter order relate to query optimization?

I'm working on a php project that's supposed to draw information from the DB for display, and I've been requested to speed up the display as much as possible.  I'm drawing data from four tables, with an additional two that I have to 'bounce' through to match the keys together.  Also, I've got five direct filtering requirements, four of the 'value = X' type and a date range.

 

My question is this: Would shuffling the placement of the filtering requirements (t1.some_key = t2.some_key and t1.some_other_value = X, etc.) make a difference in processing speed for the response time?  And if so, would the response be quicker with the direct (value = x) filters at the front of the WHERE clause, or the end?