Upon becoming Orthodox, I really didn't find anything appealing about Byzantine Chant. I, like you, felt it was mournful, and was just odd, and almost awful to my ears.
What changed me, was actually going to Greece, and experiencing that chant in church, during Holy Week and with the faithful actually belting forth the chant along with the cantors. As well as finally learning enough of the alphabet and language that I can figure out what they are saying, and what the hymns are in comparison to ours.
One of my favorite Byzantine hymns, is far more famous amongst Americans in it's Russian form. That is, "O Lord Save Thy People", or "Spasi Gospodi, iyudi tvoya", or "Soson Kyrie, ton laon sou". Once I heard that hymn in Greece, being boldly chanted by the cantors, priests & people, I was taken by it. Same for the rest of Byzantine chant.
I initially fell in love with Russian renditions of Holy Week hymns. Yet, once I heard them in Byzantine Chant, in the whole setting of a traditional, Greek church, I feel head-over-heels in love with them. Kassiani's Hymn, the Lamentation service (especially the 3 stasis), the Bridegroom hymns, They have stripped me of my garments, Today he is hung etc...
I've also fallen in love with other hymns rendered in Byzantine Chant like It is Truly Meet, To Thee, the Champion Leader, the Cherubic Hymn, God is With Us, Lord I have Cried, The Polyelos (Psalm 135) etc...
Along with all of this, sadly, I have felt that I've kind of lost my original love of later Russian Chant, while also gaining great appreciation for Gregorian & Old Roman Chant, Georgian Chant and older Russian Chants like Znamenny.
I'll post some examples here of great Byzantine Chant.
God is With Us (Isaiah)/Snami Bog/Meth'imon O TheosLord I have Cried (Psalm 140)/Kyrie Ekekraksa Pros Se/It is Truly Meet/Dostoino Est/Axion Estin
(does anyone have a rendition of this hymn in Church Slavonic, but in Byzantine Chant? I've heard it once and thought it was amazing, but can't find audio.)To Thee, The Champion Leader/Vzbrannoy voyevode/Ti Ypermaxo StratigoO Give Thanks Unto the Lord...For His Mercy Endures Forever... (Psalm 135, Polyelos)/Eksolomogisthe Tou KyriouO Lord, Save Thy People/Spasi Gospodi, ijudi tvoja/Soson Kyrie ton laon sou
It is extremely impressive to hear an entire parish singing this in Byzantine Chant. But it was made most famous outside Orthodoxy through it's use in the 1812 Overture (beginning and end): http://youtu.be/VbxgYlcNxE8Bless the Lord O My Soul/First Antiphon (Psalm 102)
Unfortunately, this isn't heard much in Greece outside of Mt. Athos because most in the Byzantine Tradition have replaced the traditional Psalms as the first three antiphons with other psalm verses alternated with petitions to Christ & the Theotokos.They Have Stripped Me/Eksedysan me ta imatia mouHoly Friday Lamentations - 1st Stasis (In the tomb they laid thee)/I Zoe en tafo