One risk with flash drives is that they can serve as a vector for the spread of malware (e.g., viruses).
Here's how malware facilitates the spread of malware. If a clean flash drive is inserted into a compromised machine, the compromised machine might write malware onto the flash drive. If the now-infected flash drive is subsequently inserted into a second machine, the malicious data on the flash drive might exploit a security vulnerability on the second computer and cause the second computer to become compromised. Flash drives may also spread malware through social engineering (AngelinaJolie.jpg, which actually contains a .exe that when clicked on launches malware). This risk is especially pronounced if you use a flash drive that you obtained from a stranger or outsider.
One possible way to partially mitigate this risk is to only use removable media internally -- never allow anyone to insert a flash drive obtained from any external source, or while travelling. However, this policy may be harder to communicate, police, and ensure compliance with than a simpler "flash drives are banned" policy.
Another possible way to partially mitigate the risk is to use write-once media (e.g., CD-Rs, DVD-Rs) to transfer large files between machines. If malicious data is stored onto the media, the write-once nature won't help you. But the way that write-once media helps is that it removes the potential that inserting clean media into a compromised machine could infect the media. Thus, write-once media is not a panacea, but it might be a way to somewhat reduce the risk.
A third possibility is to encourage people to use networked solutions for sharing files, e.g., through a secure website (secured with HTTPS and with anti-virus scanning for all uploaded files) or through secure email (with anti-virus scanning on your email gateway).