Linux + find word/string in file under directory

Actually, find already has this functionality:

find . -newermt "2013-01-01 00:00:00" ! -newermt "2013-01-02 00:00:00"

From the manpage:

-newerXY reference
       Compares  the timestamp of the current file with reference.  The
       reference argument is normally the name of a file  (and  one  of
       its  timestamps is used for the comparison) but it may also be a
       string describing an absolute time.  X and  Y  are  placeholders
       for other letters, and these letters select which time belonging
       to how reference is used for the comparison.

       a   The access time of the file reference
       B   The birth time of the file reference
       c   The inode status change time of reference
       m   The modification time of the file reference
       t   reference is interpreted directly as a time

       Some combinations are invalid; for example, it is invalid for  X
       to  be t.  Some combinations are not implemented on all systems;
       for example B is not supported on all systems.  If an invalid or
       unsupported  combination  of  XY  is  specified,  a  fatal error
       results.  Time specifications are interpreted as for  the  argu‐
       ment  to the -d option of GNU date.  If you try to use the birth
       time of a reference file, and the birth time  cannot  be  deter‐
       mined,  a  fatal  error  message results.  If you specify a test
       which refers to the birth time of  files  being  examined,  this
       test will fail for any files where the birth time is unknown.

The fastest I can come up with is to use xargs to share the load:

find . -type f -print0  | xargs -0 grep -Fil "mypattern" 

Running some benchmarks on a directory containing 3631 files:

$ time find . -type f -exec grep -l -i "mystring" {} 2>/dev/null \;

real    0m15.012s
user    0m4.876s
sys     0m1.876s

$ time find . -type f -exec grep -Fli "mystring" {} 2>/dev/null \;

real    0m13.982s
user    0m4.328s
sys     0m1.592s


$ time find . -type f -print0  | xargs -0 grep -Fil "mystring" >/dev/null 

real    0m3.565s
user    0m3.508s
sys     0m0.052s

Your other options would be to streamline either by limiting the file list using find:

   -executable
          Matches files which are executable and  direc‐
          tories  which  are  searchable (in a file name
          resolution sense).  
   -writable
          Matches files which are writable.             

   -mtime n
          File's  data was last modified n*24 hours ago.
          See the comments for -atime to understand  how
          rounding  affects  the  interpretation of file
          modification times.
   -group gname
          File  belongs to group gname (numeric group ID
          allowed).
   -perm /mode
          Any  of  the  permission bits mode are set for
          the file.  Symbolic modes are accepted in this
          form.  You must specify `u', `g' or `o' if you
          use a symbolic mode. 
   -size n[cwbkMG]  <-- you can set a minimum or maximum size
          File uses n units  of  space.  

Or by tweaking grep:

You are already using grep's -l option which cause the file name to be printed and, more importantly, stops at the first match:

   -l, --files-with-matches
       Suppress normal output; instead print the name of each input file  from
       which  output would normally have been printed.  The scanning will stop
       on the first match.  (-l is specified by POSIX.)

The only other thing I can think of to speed things up would be to make sure your pattern is not interpreted as a regex (as suggested by @suspectus) by using the -F option.