Connecting to a linux database is absolutely no different from connecting to a database on Windows, or any other platform for that matter. The interconnect protocol (SQL*net) is the same everywhere. The platforms for the client and server play no role here, neither does the fact that one is 32-bit and the other is 64-bit. Also the versions of the Oracle client and the Oracle server play little role: the protocol exchange is such that it will automatically adapt to the versions at both ends. So interconnection works with all reasonably recent versions of Oracle (say 10gR2, 11gR2, 12cR1) but they should also work with older versions (9i, even 8i). I have seen some incompatibilities reported between a 12c client and an 8.0 server ... but such is life.
What probably happens is that your Windows database runs on the same server as your ArcGIS 10 server as opposed thatwhereas your Linux database that runs on a different server. In one case you use possibly a direct connection (no IP) or maybe a local connection (localhost) where, whereas the connection to the Linux database is obviously remote and needs the full connection details: server name or IP address, TNS listener port, database service name or SID (and of course the proper credentials).
For a start it would really help if you tell us the exact error you are getting and the connection details you specify (server, port, service names). Connecting to an Oracle Database can fail for a numbervariety of reasons, but they always have to do with some mis-configuration, either client or server side.
Debugging those issues can be very difficult when using client tools (like ArcGIS) that sometimes obfuscate the error messages receivedthey receive into some generic "connection failure" message, leaving you guessing and poking.
Debugging is best done in steps, using simple tools:
VerifyConfirm that the server is reachable:
ping <database_server>
VerifyConfirm that the TNS listener is reachable on the port you expect, and knowknows about your database:
tnsping <database_server>:/<database_service>
for example:
tnsping db_server.acme.com:1521/orcl121.local
VerifyConfirm that the database is up and running and connected to your listener, and also that you are using the proper credentials.
sqlplus /@<database_server>:/<database_service>
If you are not familiar with those techniques, I recommend you seek assistance from your database administrator.
BTW, what version of Oracle are we talking about ? On Windows ? On Linux ?