Yes a popular cycleway the bit from Ringwood to Ashley Heath, is used by myself often, as the MVR in my user name is the Moors Valley Railway.

 
There is also the Moors Valley Railway

http://www.moorsvalleyrailway.co.uk/pages/track-layout not far from the old line. This was known as Castleman's Corkscrew for its indirect route through this area on its way to Dorchester.
Further reading!
Brian Jackson - Castleman's Corkscrew: Nineteenth Century Volume 1: Including the Railways of Bournemouth and Associated Lines (Oakwood Library of Railway History) Paperback 2007
Brian Jackson - Castleman's Corkscrew: Twentieth Century and Beyond Volume 2: Including the Railways of Bournemouth and Associated Lines:Paperback 2008
 Cox - Castleman's Corkscrew: Southampton and Dorchester Railway, 1844-48 City of Southampton Paperback  1975

 
The route Brockenhurst - Ringwood - West Moors - Wimborne - Poole is now a very nice cycle route. There is still evidence of Holmsley station, Ashley Heath halt, and some buffers at West Moors.

 
There has been coverage of the late and very much lamented Somerset and Dorset line via Radstock, Templecombe and Blandford Forum to Poole and Bournemouth, but little comment on the Salisbury and Wimbourne to Poole and Bournemouth line, which also survived into the 1960s.  Once both lines were closed, access by train to Bournemouth and Poole from the North, and from Devon and Cornwall, has been circuitous in the extreme - passengers having to Zig via Southampton and head in from the east, or Zag via Dorchester and head in from the west.

Here's the 1960 timetable for the Salisbury to Poole and Bournemouth service a handful of years before it closed.  It's ironic that with the Western Gateway subNational Transport body, this line would have provided a link (the only link) between the two sections of rail transport to be overseen by that new transport body.