What tools did you use to draw the logo? Flash drawing tools produce vector elements which will resize without pixelating/blurring like you would get with bitmaps (jpegs/photos/etc).
It sounds like you exported the image and then did the resize. Instead, go back to Flash and export the image again, say as PNG, and when you do that you're even a dialog box where you can type in the width and height. Type in a much bigger number (you can work out what size would make sense, based on the DPI of the printer that will make the teeshirt). For example, if the logo is to be 6 inches across, and the printer can do 300 DPI, the width of the PNG should be 1800.