I know it shouldn’t matter, but for me it sometimes does. I get annoyed with the blonde-haired, blue-eyed Jesus that is so often depicted, and I was happy to see Mel Gipson made sure to color Jesus’ eyes as brown in The Passion. Jesus is our Lord first and foremost, but he was also a man, a Middle-Eastern Jew, and I feel that that’s very important to his human identity. Jesus came from that region, from that people, for a reason, and casually side-stepping his actual ethnicity does a disservice to the wholeness of Scripture, IMO.
I imagine Jesus looked very similar to His modern-day relatives, the Palestinian Arabs and the Mizrahi Jews: tan to light brown skin, very dark, wavy hair, and deep, dark eyes. Sometimes it’s hard to keep this image of Him in my mind because of the perpetuation of other depictions, but I try to keep that focused as much as possible. Jesus was a Jew, a man from Middle-East, and His lineage was very important to His person and His mission of Salvation.
EDIT:
Ironically, many people from the Caucasus mountains, like my mother’s family, are dark-skinned and share many features in common with Persians (Iranians).