Actually, everything matters. It's why I just buy coils...
The two biggest things that matter for the specifications you've given are \$A_L\$, which determines the inductance per turns\$^2\$, and the point at which the core saturates.
If I'm remembering all this stuff correctly, the core will saturate at a certain magnetic field strength, depending on the material. The field strength depends on the coil current times turns, divided by the average magnetic path through the core. So lots of turns on an itty-bitty core will get you the inductance you want, but the core may saturate. You need to choose a core large enough to get you the inductance you want without saturating the core.
While you're stressing out over that, you also need to consider that the core is also lossy, both because it's (possibly) conductive and because reversals in the magnetic field dissipate energy in the core (if you look at the B-H curve of a material, the area inside that curve is proportional to the energy lost each cycle).
And, finally, the wire you use is also lossy, and you can only go so big (and in high-frequency applications, it works better to use individually insulated multiple strands, to get a sorta-litz wire effect).
If you check out the Fair-rite website, they have a ton of instructional videos. Your cores are probably Fair-rite, or Fair-rite knockoffs, so the videos should be helpful even down to the material designations.