I'm trying to use this program in the first place just as a basic midi multi-tracker as it's been recommended here by a few people. (my old program, MSI, has suddenly developed incompatibility with Windows 7 - v frustrating)

But there's an immediate problem with Reaper, which is that as soon as I start monitoring the DP, there is a delay of around 700ms, making it pretty useless for anything - and entirely useless for multi-tracking. This cannot be right, yet I've searched through the on-line manual for a solution but found nothing.

My other, supposedly inferior program never had this problem, and worked well on default settings for monitoring live & recorded MIDI tracks.

Anyone know how to solve this with Reaper, or another reasonably priced midi program which won't have this sort of delay?

What do You mean by "monitoring"?
Are You using MIDI In/Out in Reaper configured to use only Your DP or You have some VSTi in Reaper?

Usually such big delays in MIDI sequencers are caused by setting the MIDI Out device to some soft-synth, especially when the sound card isn't ASIO capable and is using standard Windows drivers.
Or even when the ASIO setup is correct, but some VST plugins used don't compensate the processing delay.
Tell us more details about Your setup.

No, that would be more understandable, as you say, if the DP were being used to trigger another VST sound source. But no, I'm only using the DP's own sound engine, so the computer is receiving the midi signal, it's being recorded in Reaper and playing back in the DP.

But there is this massive 0.7 second delay when it's looped through Reaper. This does not happen in the other virtual multitrackers I've used. But Reaper does look like a good program apart from this.

Then maybe You should try to search on Reaper's forums.
For example http://forum.cockos.com/showthread.php?t=88953

Thank you very much, Kumi 27. This problem is fixed now - the link to the Reaper forum is was very useful. Now seems a good time to jump ship to Reaper from Frieve MSI, which I've been using but is totally unsupported (no entries on its website since 2008!).