First, I think the bitdepth can be a problem. Are you using A8R8G8B8? I think this can be a little insufficient on older cards. Try to use a lower bitdepth for your textures.

And yes, it is always a good advice to use textures sized power of 2!

256*256 is the fastest and you can tile it in several patterns if needed (for animations or backgroundtiles. But don't use graphics which are stored in size like 800*600 or 100*123.

The Z-Order thing is another point you should take a look on. The order you are drawing things can extremely slow down (or speed up) rendering.

Other things you could take a look at are:
- Screen resolution (older cards are faster at lower resolutions)
- Are you using Antialiasing? Is this supported by the hardware?
- blending with alphachannel is hardware intensive work for the video card. The more you use it, the slower rendering will be. Try to avoid it where it is possible
- Every DirectX functionality you are using, which is not supported by the hardware will be executed in some kind of software emulation which slows down everything. Take a closer look at your drawing routines.

Can you provide a screenshot so that I can imagine of what "complexity" your renderings are?

And finally: Why are you using DelphiX instead of Asphyre eXtreme?