You can't 1031 into a regular REIT, according to what I was reading. You can 1031 into an UPReit (different entity.)
This is a big gotcha, though:
"Risks Associated With A 1031 Exchange Into An UPREIT
Investors who 1031 Exchange into an upREIT have exchanged into a security and therefore no longer own real estate. Since the investor now owns a security, he or she cannot 1031 Exchange out of the upREIT and into other real estate. The sale or disposition of their interest in an upREIT will result in a taxable transaction, including the recognition of their deferred capital gain and any depreciation recapture.
The upREIT also has control over the asset they 1031 Exchanged into and therefore has control over the sale or disposition of the asset. The sale or disposition of the asset can trigger the recognition of the investors deferred capital gain and any depreciation recapture. Some upREIT sponsors will guarantee that they will not trigger any taxable gain for a specified number of years, while others remain silent regarding the potential for triggering the deferred taxable gain."
--50.82.xxx.xx