The GNU Extension to the GNU Fortran compiler provides the subroutine GETCWD() that well, gets the current working directory. However, my code has to be portable to the ifort and nagfor compiler as well and I use F2003 features.
So, is there an alternative to GETCWD() for F2003 and later?
I have the standard here but it's quite sizeable and I've been going through it for a while now and haven't found anything useful...
As noted in the comments, you can make use of get_environment_variable which is standard Fortran (e.g. F2008 13.7.67).  This example program queries the value of $PWD, which should contain the directory your shell is in when you invoked the executable.
program test
 implicit none
 character(len=128) :: pwd
 call get_environment_variable('PWD',pwd)
 print *, "The current working directory is: ",trim(pwd)
end program
And its output:
casey@convect code % pwd
/home/casey/code
casey@convect code % so/getpwd 
 The current working directory is: /home/casey/code
This is standard Fortran, but its portability will be limited to Unix and Unix-like shells that set this variable.
Another option while standard but ugly (in my opinion) would be to use execute_command_line to run a command that can output the working directory to a temporary file (e.g. pwd > /tmp/mypwd), then reading that file. 
You can also use the ISO_C_Binding and call the corresponding C functions: 
cwd.c:
#ifdef _WIN32
/* Windows */
#include <direct.h>
#define GETCWD _getcwd
#else
/* Unix */
#include <unistd.h>
#define GETCWD getcwd
#endif
void getCurrentWorkDir( char *str, int *stat )
{
  if ( GETCWD(str, sizeof(str)) == str ) {
    *stat = 0;
  } else {
    *stat = 1;
  }
}
test.F90:
program test
 use ISO_C_Binding, only: C_CHAR, C_INT
 interface
   subroutine getCurrentWorkDir(str, stat) bind(C, name="getCurrentWorkDir")
     use ISO_C_Binding, only: C_CHAR, C_INT
     character(kind=C_CHAR),intent(out) :: str(*)
     integer(C_INT),intent(out)         :: stat
    end subroutine
  end interface
  character(len=30)   :: str
  integer(C_INT)      :: stat
  str=''
  call getCurrentWorkDir(str, stat)
  print *, stat, trim(str)
end program
This code is valid for Windows and Unix-derivates (Linux, OSX, BSD, etc. )
The accepted answer contains two errors (it passes the wrong value as the length of the string to GETCWD, and leaves in the C_NULL_CHAR). This answer corrects those mistakes and makes the interface more usable from Fortran.
The basic idea is the same: call getcwd or _getcwd using C, and call the C wrapper using Fortran's C interoperability features. On the Fortran side, a wrapper subroutine is used to handle the string length, so it does not have to be passed explicitly.
Also, C_INT and C_CHAR are not necessary the same as default integers and default characters, which are wanted on the Fortran side (though in practice I am not aware of any system where C_CHAR and default char differ). The wrapper also converts those. Also, the string returned from C contains the terminating C_NULL_CHAR, which must be removed for the string to be usable on the Fortran side.
The C code:
#ifdef _WIN32
#include <direct.h>
#define GETCWD _getcwd
#else
#include <unistd.h>
#define GETCWD getcwd
#endif
/* Return 0 on success, 1 on error. */
int getCWDHelper(char *str, int len)
{
    return GETCWD(str, len) != str;
}
The Fortran code:
module cwd
    use iso_c_binding, only: C_INT, C_CHAR, C_NULL_CHAR
    implicit none
    private
    public :: getCWD
    interface
        function getCWDHelper(str, len) bind(C, name="getCWDHelper")
            use iso_c_binding, only: C_INT, C_CHAR
            integer(kind=C_INT) :: getCWDHelper
            character(kind=C_CHAR), intent(out) :: str(*)
            integer(kind=C_INT), value :: len
        end function getCWDHelper
    end interface
contains
    ! Writes the current working directory path into str.
    ! Returns 0 on success, or 1 on error.
    function getCWD(str)
        integer :: getCWD
        character(*), intent(out) :: str
        integer :: i, length
        character(len=len(str), kind=C_CHAR) :: str_copy
        ! Call the C helper, passing the length as the correct int kind
        getCWD = getCWDHelper(str_copy, len(str_copy, kind=C_INT))
        if (getCWD /= 0) then
            str = '' ! Error, clear the string
            return
        end if
        ! Copy the C_CHAR string to the output string,
        ! removing the C_NULL_CHAR and clearing the rest.
        length = index(str_copy, C_NULL_CHAR) - 1
        do i = 1, length
            str(i:i) = char(ichar(str_copy(i:i)))
        end do
        str(length+1:) = ''
    end function getCWD
end module
Test code:
program test
    use cwd, only: getCWD
    implicit none
    character(len=255) :: path
    integer :: error
    error = getCWD(path)
    print *, error
    if (error == 0) print *, path
end program
Making the return value allocatable and looping to get a sufficient size is left as an exercise to the reader.
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