Norse myth associates the east with evil...
From the Voluspa, 1999 Larrington translation:
From the east falls, from poison valleys
a river of knives and swords, Cutting it is called
In the east sat an old woman in Iron-wood
and nurtured there offspring of Fenrir;
a certain one of them in monstrous form
will be the snatcher of the moon
Hrym drives from the east, he has his shield before him,
the great serpent writhes in giant rage ...
(stanza 50; Hrym is a giant and captain of Naglfar)
A ship journeys from the east, Muspell's people are coming
over the waves, and Loki steers.
There are the monstrous brood with all the raveners,
The brother of Byleist is in company with them.
(stanza 51, Byleist is brother of Loki; this last line is a kenning, a common poetic device in eddaic and skaldic poetry)
It should be noted, however, that as per stanza 52, "Surt comes from the south" (the stanzas seem to be misnumbered in my edition or something, it goes stanzas 47, 49, 50, 51, 48, 52, and then carries on up from 52 as normal).
Anyway, evil does primarily seem to come from the east in Norse myth.
...but not many other types of myth seem to.
In Finnish myth, Louhi is a wicked queen of the land known as Pohjola; Pohjola is located in the far north; to quote the Kalevala: "In these dismal Northern regions / In the dreary land of Pohja". To quote wikipedia here:
the idea of an otherworldly far north is a widespread motif in both Classical and medieval European literature
There seems to be, at the minimum, no association of the east with the evil in Finnish myth as far as I can tell.
Greek myth was actually influenced a bit by the east, but the closest thing I could find to an association of evil with the east was Dionysus fighting all the way to India and then getting stopped there, but that had a sort of positive light, as it was a weird drunken revelry war thing. Dionysus is weird. Anyway, interestingly, as you move into Roman myth (which is obviously closely associated with Greek myth) it incorporates stuff from Eastern mythology:
For instance, the cult of Sun was introduced in Rome after Aurelian's successful campaigns in Syria. The Asiatic divinities Mithras (that is to say, the Sun) and Ba'al were combined with Apollo and Helios into one Sol Invictus, with conglomerated rites and compound attributes.
So not a negative connotation at all.
Tolkien influence explains this
Further, I can see very little beyond Norse association of the east with evil. This does make some sense wrt Tolkien though, because he was heavily influenced by Norse myth, and was one of his predominant studies. It should be noted that Tolkien was also heavily influenced by the Kalevala especially wrt the story of Turin Turambar. Another note, though, to quote Tolkien's letter 229:
The placing of Mordor in the east was due to simple narrative and geographical necessity, within my 'mythology'. The original stronghold of Evil [Thangorodrim] was (as traditionally) in the North; but as that had been destroyed, and was indeed under the sea, there had to be a new stronghold, far removed from the Valar, the Elves, and the sea-power of Númenor.
See also this sci fi and fantasy question.