I have the latest UBB.threads (7.5.9) running on PHP 5.6 without any problems; what builds are available on your new system? I'd recommend PHP v5.6 as it's stable (unlike PHP7) and it is a lot faster than PHP v5.3. Also, keep in mind that UBB.threads v7.5.8 was primarily pushed out for PHP compatibility, and includes some security fixes as well.
[root@server ~]# php -v
PHP 5.6.25 (cli) (built: Aug 20 2016 09:39:48)
Note that UBB.threads v7.6.0 will
REQUIRE PHP 5.3+, you can grab the test tool at UBBWiki:
UBB.threads v7.6.0 System Requirement Test 20160222. The installer/upgrade tool will die graciously and present a warning for users not running PHP v5.3+.