Following from Mr. Wizard's post, I was able to get Mathematica to substitute Greek letters with whatever font I want by editing
\SystemFiles\FrontEnd\TextResources\UnicodeFontMapping.tr. I first tried changing
Mathematica: (Mathematica1 Mathematica1Mono) Automatic
Mathematica: (DejaVuSans DejaVuSansMono) Automatic
which worked for changing the font, however it forced all of the symbols in Mathematica1 to be rendered with DejaVu - including things DejaVu obviously doesn't have like the line for a fraction and the \[Wolf] symbol.
So I undid that, and then just changed the Greek letters to reference font set 0 instead of 2, -2, or 4, which works so long as you use a font that supports a wide range of Greek letters.
0x0391 N 2 0x41 # \[CapitalAlpha]
0x0391 N 0 0x41 # \[CapitalAlpha]
I did this through the Sampi symbol, and now the Greek looks much better with any font. Here's my file if you want to copy it.
Here's what it will look like if you do these changes:
The first two are what I want Mathematica to do - display all Greek in whatever font I say just like any other text. The Text cell is in DejaVuSans, and the Input cell is in DejaVuSans Mono Plain.
The next two are what happens if you change your Unicode Mapping but don't change the default font - for Text it's Times and Input it's Courier Bolded. Notice that they're both missing some of the fancier symbols.
Finally, I have the Mathematica1 and Mathematica1Mono fonts. If you do not change your Unicode Mapping, this is what you get, Mathematica1 for Plain text, and Mathematica1 for mono spaced text. Ignore that some of the symbols are wrong here, that's my own fault.