Go in to settings and down to "About Phone" or "Software Updates", each phone can have this in a different area of Settings. Check for firmware or ROM updates, if there are any get them, if not it probably is a flash issue.
Also there is a Browser that you can download that will enable flash on some phones that are capable of handling it but won't play, called SkyFire.
A friend of mine has the I Pad 1 which will not play flash by default, with SkyFire his IP will play FLV videos. Because of Apples dislike for flash and the fact that so much video on the web is flash, he uses SF only and forgoes the default Apple browser all together. While this example is on an I Pad it holds true for Android products as well, the browsers built in flash capabilities by passes the phones/tablets issues.
Note that this works on many but not all hardware so if all else fails, give it a try.