After getting my steel frame powder coated, I noticed that some sand from the media blasting had collected on some grease inside the frame. I definitely don't want to install the bottom bracket without first cleaning this.. How should I go about doing this without damaging the new powdercoat and not rust the inside of my frame?

I would run a bottom bracket tap through, and consider facing the bb. All of this should've been masked off before the powdercoat...

Clean what you can reach with a rag. For the seat tube, make a small wad, and push it through with a stick. Down tube and chain stays tie a string to a corner (securely) push the rag up the tube with a coat hanger, then pull it back with the string.

When you've gotten the worst out, spray some WD-40, or a light solvent/oil mix (10:1) all around and let it drain to the BB overnight and wipe out the residue. Odds are that over time a bit more grit may fall down from places you didn't reach, but unless you use a loose ball bottom bracket withour a plastic liner it won't matter.

Good idea, but not until the loose grit has been cleaned out. The abrasive grit will wreak havoc on those expensive cutting tools. If there's powder coat on e.g. the bottom bracket faces, use a file to score it so the facer tool can get some traction.

The easiest way to remove the grit is to blow compressed air through the frame.

Would it be unwise to flush with water if the weather is warm?

No, but it probably won't do you a ton of good if the grit is stuck in an oil or grease film.

If you want to use water, mix up a batch of strong dishing washing soap and water, and flush with that or use a rag soaked it it. When finished drain the frame and dry completely in a warm place.

I find a shotgun cleaning rod with a 10-gauge brass bristle brush works well to clean the inside of seattubes. You can wrap it in cloth to increase the diameter if needed and use either solvent or strong detergent to clean out the grit and grease. there are flexible cleaning rods that could be used to clean the inside of the frame tube you don't have straight access to.