Suppose you have a transmission line (made with copper) with a
rectangular cross section oriented along z-axis. People talk about the
wave characteristics such as reflection in the line. So in this case I
assume the wave is propagating in z direction. Therefore, the electric
field should be in transverse plane. Since the current density vector
J = sigma*E, then the current should flow parallel to xy plane. But in
the circuit point of view, the current shuld flow along the
transmission line (z axis). So I am very confused about the direction
of the current. I am very sure that I have misunderstood so many
things. So I will really appreciate if you could clarify this case.

Thank you

"transmission line" is an abstract concept, which can be applied to many
 physical implementations: waveguide 

http://www.microwaves101.com/encyclopedia/waveguideconstruction.cfm

coaxial cable, optical fiber or even an hydraulic pipe...

The main criterion is that the basic 'four terminal model'
http://en.wikipedia.org/wiki/Transmission_line
with appropriate matrix calculus is applicable.

I the case where the wave in question is electromagnetic,
 the direction of the relevant vectors (E,H,j ..)
 may depend on the details not given in you question.

Nevertless, this may help : The equation for current you mention
- continuum form of Ohm's Law -
http://en.wikipedia.org/wiki/Ohm's_law
may play some role in a lossy transmission, but does not represent 
the propagation of energy, which (e.g. in a waveguide,..) 
is same as in free emg wave:
http://www.monos.leidenuniv.nl/smo/index.html?basics/light.htm
It is given by Pointing vector which is E x H - that is perependicular to E.
Ohmic current, parallel to E, dissipates, rather then propagates the energy.

OK?

Playing with these applets may help
http://www.amanogawa.com/index.html



Hedgie

The answer provided by the researcher is not helpful for me in any
case. The answer never clarify the issues I raised. He just pointed
out some links which also are very general to my question and could
not help me. I listed this question for $15 for a detail answer. But
now I feel the answer provided is not worth even for $2. So please
consider this situation and refund the amount.