I also grew up in Oceanway and it's not bad at all, their are some scumbags in the area, like anywhere else. As for bad areas, I'll list a few from areas I'm familiar with.
Northside:
Dunn Ave. from about i-95 to Lem Turner, it's a mix, but has some bad areas if you leave the main road, Harts Rd. is the worst.
Lem Turner from Dunn Ave. South all the way to well, ALL THE WAY...is pretty much horrible, it leads you straight into HELL (Northwest side).
Oceanway has a couple of areas with some thieving pieces of trash. Kaleel Rd. off of Farm Rd. in the Dunn Creek area had a whole group of shitheads that liked to steal alot living there, I'm sure they are still there. The one thing about Oceanway is the growth it is experiencing, some of the folks around there aren't aware that segregation is over.
Arlington: Justina area. Pretty much every apartment complex over there is a shit hole.
Beaches: Mayport, Jacksonville Beach (south of Beach Blvd. just off of A1A).
Southside: Philips Hwy from the north end to pretty much University Blvd. is crap. Particulary Emerson St. and Philips/St. Augustine Rd. area.
Mandarin: A mix in my opinion. Pretty Good area overall. Most of the Apartments on Sunbeam have problems. This area seems to attract criminals, they go there to steal.
Westside/Northwest Side: Lane Ave./Miss Muffet Area. 103rd./Old Middleburg area., Cassat/San Juan area, Lenox/Lake Shore area. LABELLE is a bad one, runs off Cassat.
The northwest side I-10 west of the river and north to Dunn Ave is the worst part of Jacksonville IMHO. That will cover where most of your murders are occuring.
Springfield(Downtown): Main, from downtown north to the river. Don't venture down there unless you have to.
All in all if you stay on the Main Roads and dont venture off into any neighborhoods you aren't familiar with, you'll be ok. The city is overcrowded so it's got it's problems obviously, which is why I live in Baker County and work in Jax.