[Verizon Galaxy Note 3 Q&A, Help & Troubleshooting] ronjwright:
Apparently, I have the same issue, too. I cleared the caches of the Email and Exchange services apps and restarted Exchange services, which didn't work out. I even checked the Android issue tracker, but I couldn't find anything.

EDIT: I discovered that an exception occurs after navigating to the sent folder. Here is the logcat output:

W/System.err( 5864): android.content.pm.PackageManager$NameNotFoundException: com.samsung.helphub
W/System.err( 5864):	at android.app.ApplicationPackageManager.getPackageInfo(ApplicationPackageManager.java:104)
W/System.err( 5864):	at com.android.email.activity.MessageListXL.onPrepareOptionsMenu(MessageListXL.java:4790)
W/System.err( 5864):	at android.app.Activity.onPreparePanel(Activity.java:2606)
W/System.err( 5864):	at com.android.internal.policy.impl.PhoneWindow.preparePanel(PhoneWindow.java)
W/System.err( 5864):	at com.android.internal.policy.impl.PhoneWindow.doInvalidatePanelMenu(PhoneWindow.java)
W/System.err( 5864):	at com.android.internal.policy.impl.PhoneWindow$1.run(PhoneWindow.java)
W/System.err( 5864):	at android.view.Choreographer$CallbackRecord.run(Choreographer.java:803)
W/System.err( 5864):	at android.view.Choreographer.doCallbacks(Choreographer.java:603)
W/System.err( 5864):	at android.view.Choreographer.doFrame(Choreographer.java:572)
W/System.err( 5864):	at android.view.Choreographer$FrameDisplayEventReceiver.run(Choreographer.java:789)
W/System.err( 5864):	at android.os.Handler.handleCallback(Handler.java:733)
W/System.err( 5864):	at android.os.Handler.dispatchMessage(Handler.java:95)
W/System.err( 5864):	at android.os.Looper.loop(Looper.java:157)
W/System.err( 5864):	at android.app.ActivityThread.main(ActivityThread.java:5293)
W/System.err( 5864):	at java.lang.reflect.Method.invokeNative(Native Method)
W/System.err( 5864):	at java.lang.reflect.Method.invoke(Method.java:515)
W/System.err( 5864):	at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:1265)
W/System.err( 5864):	at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:1081)
W/System.err( 5864):	at dalvik.system.NativeStart.main(Native Method)
W/System.err( 5864): android.content.pm.PackageManager$NameNotFoundException: com.samsung.helphub
W/System.err( 5864):	at android.app.ApplicationPackageManager.getPackageInfo(ApplicationPackageManager.java:104)
W/System.err( 5864):	at com.android.email.activity.MessageListXL.onPrepareOptionsMenu(MessageListXL.java:4790)
W/System.err( 5864):	at android.app.Activity.onPreparePanel(Activity.java:2606)
W/System.err( 5864):	at com.android.internal.policy.impl.PhoneWindow.preparePanel(PhoneWindow.java)
W/System.err( 5864):	at com.android.internal.policy.impl.PhoneWindow.doInvalidatePanelMenu(PhoneWindow.java)
W/System.err( 5864):	at com.android.internal.policy.impl.PhoneWindow$1.run(PhoneWindow.java)
W/System.err( 5864):	at android.view.Choreographer$CallbackRecord.run(Choreographer.java:803)
W/System.err( 5864):	at android.view.Choreographer.doCallbacks(Choreographer.java:603)
W/System.err( 5864):	at android.view.Choreographer.doFrame(Choreographer.java:572)
W/System.err( 5864):	at android.view.Choreographer$FrameDisplayEventReceiver.run(Choreographer.java:789)
W/System.err( 5864):	at android.os.Handler.handleCallback(Handler.java:733)
W/System.err( 5864):	at android.os.Handler.dispatchMessage(Handler.java:95)
W/System.err( 5864):	at android.os.Looper.loop(Looper.java:157)
W/System.err( 5864):	at android.app.ActivityThread.main(ActivityThread.java:5293)
W/System.err( 5864):	at java.lang.reflect.Method.invokeNative(Native Method)
W/System.err( 5864):	at java.lang.reflect.Method.invoke(Method.java:515)
W/System.err( 5864):	at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:1265)
W/System.err( 5864):	at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:1081)
W/System.err( 5864):	at dalvik.system.NativeStart.main(Native Method)

I'm not sure if this is related to the issue.

EDIT: Now I feel totally stupid. The reason why I wasn't seeing any of my sent messages was because I failed to realize how much time passed since I sent my last message. When I set the sync period of the Sent folder to one month, I can see all the messages I have sent in the past month. I guess this issue doesn't affect me after all, but at least I helped by showing that a trivial import of the Email app from the MJE ROM doesn't work.