I have had to use different approaches on different networks based on what the user really wants.
If the file sizes are large, remote network connections like VPN and LogMeIn will be somewhat slow and unworkable. Constantly connected programs like MYOB also tend not to like remote connections and can lead to corrupted databases.
The few times we have needed access to specialised data or programs I have ended up using the Windows XP Remote Desktop function, allowing us to keep the licensing for special programs and our databases in-house.
The rest of the time it is only email the end user really wanted, so mobile devices like Blackberry work out better. (assuming a good spam filter)
Small files like end of week timesheets and such just have them email. If you have no fixed IP and want to update small files, stick with LogMeIn.
I recommend email or a proper VPN because at some point you may want to keep a record of connections and what data was historically sent.