Abstract An OLRMTS ( v ) ( OLARMTS ( v ) ) over a ( v + 1 ) -set X is a collection { ( X ∖ { x } , B x ) : x ∈ X } of v + 1 pairwise disjoint resolvable (almost resolvable) Mendelsohn triple systems of order v . In this paper several direct construction methods for OLRMTSs and OLARMTSs are presented and then applied to produce some new orders; the smallest unknown OLRMTS ( v ) for v = 18 , 24 , the smallest unknown OLARMTS ( v ) for v = 19 , 22 , 28 , 31 , and some other small designs are displayed; a few new existence families are also obtained by known recursive constructions.