You should simply contact the person (or people) who is responsible for travel and expense reimbursements and ask them if there is anything that you can do to assist in processing your expense report. That way you won't seem to be "bugging" them about the slowness of the process and they may be able to give you some advice on how to make future reimbursements go through the system faster.
Frankly, if travel and/or expense reimbursements DO become a long-term problem with this company, you might consider other employment options. If they are unwilling to issue employees corporate cards, yet they are equally unwilling to reimburse them promptly for expenses accrued during company business or travel. that may speak volumes on how concerned they are about their workforce.