Important text
Other important text
>> Not important 1
>> Not important 2
>> Not important 3
Other important text

var allSpanTags = document.getElementsByTagName('span');

for(var thisSpanTag in allSpanTags){
   thisSpanTag.innerHTML.deleteStringBetween(index_of_first_">"_in_given_spans_inner_HTML  ,  index_of_end_of_innerHTML_of_this_span_tag);
}

var allSpans = document.getElementsByTagName("span");
for(var i = 0; i < allSpans.length; i++) {
    try{
         if (allSpans[i].className.indexOf("textblock") > -1) {
             allSpans[i].innerHTML = allSpans[i].innerHTML.replace(/>.*/g, '');
         }
    }catch(e){}
}